Abstract

An \((\alpha ,\beta )\)-ruling set of a graph \(G=(V,E)\) is a set \(R\subseteq V\) such that for any node \(v\in V\) there is a node \(u\in R\) in distance at most \(\beta \) from v and such that any two nodes in R are at distance at least \(\alpha \) from each other. The concept of ruling sets can naturally be extended to edges, i.e., a subset \(F\subseteq E\) is an \((\alpha ,\beta )\)-ruling edge set of a graph \(G=(V,E)\) if the corresponding nodes form an \((\alpha ,\beta )\)-ruling set in the line graph of G. This paper presents a simple deterministic, distributed algorithm, in the \(\mathsf {CONGEST}\) model, for computing (2, 2)-ruling edge sets in \(O(\log ^{*}n)\) rounds. Furthermore, we extend the algorithm to compute ruling sets of graphs with bounded diversity. Roughly speaking, the diversity of a graph is the maximum number of maximal cliques a vertex belongs to. We devise \((2,O(\mathcal {D}))\)-ruling sets on graphs with diversity \(\mathcal {D}\) in \(O(\mathcal {D}+\log ^{*}n)\) rounds. This also implies a fast, deterministic \((2,O( \ell ))\)-ruling edge set algorithm for hypergraphs with rank at most \( \ell \).

Furthermore, we provide a ruling set algorithm for general graphs that for any \(B\ge 2\) computes an \(\big (\alpha , \alpha \lceil \log _B n\rceil \big )\)-ruling set in \(O(\alpha \cdot B \cdot \log _B n)\) rounds in the \(\mathsf {CONGEST}\) model. The algorithm can be modified to compute a \(\big (2, \beta \big )\)-ruling set in \(O(\beta \varDelta ^{2/\beta } + \log ^{*}n)\) rounds in the \(\mathsf {CONGEST}\) model, which matches the currently best known such algorithm in the more general \(\mathsf {LOCAL}\) model.